On the feasibility of deriving pseudo-redshifts of gamma-ray bursts from two phenomenological correlations

In recent decades, GRBs have been found to exhibit various phenomenological correlations among parameters of the prompt emission. Among these are two robust correlations which correlate the spectral peak energy (in the $\nu f_{\nu}$ spectrum), $E_{\mathrm{p}}$, to isotropic energy $E_{\mathrm{iso}}$ (Amati et al., 2008), and isotropic peak luminosity $L_{\mathrm{p}}$ (Yonetoku et al., 2004), i.e.,

There exists two distinct tracks for LGRBs and SGRBs in $E_{\mathrm{p},z}-E_{\mathrm{iso}}$ (Amati) space, with SGRBs forming a parallel track above LGRBs, due to their lower energy output (See Fig.1). This may be attributed to their shorter durations. However, in $E_{\mathrm{p},z}$ – $L_{\mathrm{p},z}$ (Yonetoku) parameter space, there exists no significant separation between LGRBs and SGRBs.

The origin of these correlations remains a topic of active debate. While some have attributed them as being an artifact of instrumental selection biases (Band & Preece, 2005), several studies have quantified the impact of selection biases and concluded that, although selection effects may influence the slope and scatter, they cannot fully account for the existence of the spectral energy correlations. Further evidence for the Amati and Yonetoku correlations comes from the discovery of an $L-E_{\mathrm{p}}$ relationship that appears within individual bursts (Liang et al., 2004; Frontera et al., 2012). Notwithstanding the uncertainties in the underlying physics, many have considered their potential utility in deriving “pseudo-redshifts” (Yonetoku et al., 2004; Dainotti et al., 2011; Tan et al., 2013; Tsutsui et al., 2013; Deng et al., 2023), which would be particularly vital for population studies. For example, Zhang & Wang (2018) applied the Yonetoku relation to constrain the luminosity function and formation rate of SGRBs, while others, such as Wanderman & Piran (2015) and Paul (2018), have investigated the evolution of the luminosity function in depth. Given that Fermi-GBM has amassed a dataset of over 2,400 GRB detections since its launch (Poolakkil et al., 2021), we believe it is pertinent to evaluate the viability of using these correlations for deriving pseudo-redshifts.

In this study, we test the feasibility of using the Amati and Yonetoku relations to analytically derive pseudo-redshifts by utilizing a synthetic catalogue of GRBs and comparing the inferred psuedo-redshifts (henceforth denoted as $z_{\mathrm{i}}$) to their true, generated redshift $z_{\mathrm{g}}$. We mainly consider the case for LGRBs as they have traditionally been used for cosmological studies; however, we also provide some analysis for SGRBs in the Discussion. The paper is organized as follows: section 2 describes our detailed simulation process; in section 3, we summarize our results, and section 4 offers a summary and discussion.

The Amati correlation is defined in terms of two intrinsic (i.e., rest-frame) quantities: the peak energy $E_{\mathrm{p},z}$ and the isotropic energy $E_{\mathrm{iso}}$, for a given redshift $z$. The intrinsic parameters $E_{\mathrm{p},z}$ and $E_{\mathrm{iso}}$ are related to their observed counterparts through

and

where $F$ is the fluence, $k$ is the k-correction (which we take as $k=1$, due to the wide energy coverage of Fermi-GBM), and $D_{\mathrm{L}}$ is the luminosity distance, i.e.,

We adopt the Planck18 cosmology, assuming a flat $\Lambda$CDM universe with $H_{0}=67.66\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$ and $\Omega_{\mathrm{M}}=0.3111$ as reported by Aghanim et al. (2020). Holding $E_{\mathrm{p,o}}$ and $F$ fixed for a given burst, and allowing $z$ to vary continuously (over $z\in[0.1,10]$), we trace out a continuous curve in the $E_{\mathrm{iso}}-E_{\mathrm{p},z}$ plane. We refer to this locus of points as

Geometrically, it is a parametric curve where the running parameter is $z$. We next compare $\mathcal{A}(z;E_{\mathrm{p,o}},F)$ to the best-fit Amati line; any intersection between $\mathcal{A}(z;E_{\mathrm{p,o}},F)$ and this correlation line corresponds to a candidate “pseudo-redshift” $z_{i}$: that is, a redshift for which the observed $(F,E_{\mathrm{p,o}})$ values place the burst exactly on the Amati relation.

We fit the Amati parameters $(a_{\rm A},b_{\rm A})$ to the LGRB sample of Lan et al. (2023), finding $a_{\rm A}=0.46\pm 0.016$ and $b_{\rm A}=-21.84\pm 0.87.$ We first simulate a population of 100 GRBs that perfectly obey this best-fit Amati line ¹¹1All the relevant codes in this paper can be accessed from the GitLab repository </> . Specifically, we first draw $E_{\mathrm{iso}}$ values from the observed distribution of Lan et al. (2023) and sample the “true” redshift $z_{\mathrm{g}}$ for each GRB from a typical LGRB formation rate $\psi(z)$ (Salvaterra & Chincarini, 2007):

where $\Sigma(z)$ is the metallicity-convolved efficiency function (Kewley & Kobulnicky, 2005)

$\psi_{*}(z)$ is the star formation rate from Li (2008) and we take the normalization factor $k_{\rm GRB}=1.0$. For each simulated burst, we set

and then compute its observed fluence and observed peak energy via Equations. 3 and 4.

Thus, each simulated GRB is assigned both intrinsic parameters $\{E_{\mathrm{iso}},\,E_{\mathrm{p},z_{\rm g}},\,z_{\mathrm{g}}\}$ and observed parameters $\{F,\,E_{\mathrm{p,o}}\}$. Finally, to infer its pseudo-redshift, we construct the parametric curve $\mathcal{A}(z;E_{\mathrm{p,o}},F)$ and find any intersection(s) with the Amati line. The resulting solution(s) $z_{i}$ can then be compared to the known “true” $z_{\mathrm{g}}$. Reliable distance indicators must yield only a single redshift solution within a reasonable redshift range, and so we wish to see if the Amati relation can yield a unique solution $z_{i}$ for each GRB, given its observed spectral peak energy $E_{\mathrm{p,o}}$ and fluence $F$. Next, we add intrinsic scatter to the simulated data, based on our measured dispersion of $\sigma_{\log E_{{\rm p},z}}$ = 0.30 on the data of Lan et al. (2023), where we attempt to gauge the statistics of $z_{\rm i}$ with a simulated catalogue of 500 GRBs; this dispersion defines an upper and lower boundary around the best-fit Amati relation, forming the uncertainty region $\mathcal{U}_{\mathrm{A}}$. For each simulated GRB, we determine its 1$\sigma$ redshift range $\bigl{[}z_{\mathrm{i}}^{-\sigma},\,z_{\mathrm{i}}^{+\sigma}\bigr{]}$ by finding where the parametric curve $\mathcal{A}(z;E_{\mathrm{p,o}},F)$ intersects $\mathcal{U}_{\mathrm{A}}$.

In the case of the Yonetoku relation, the intrinsic luminosity is given by

where $f_{\gamma}$ is the observed flux. Since we are substituting $E_{\rm iso}$ with luminosity, the GRB parametric curve $\mathcal{Y}(z;E_{\mathrm{p,o}},f_{\gamma})$ in Yonetoku parameter space takes on a distinct functional form with respect to $z$ than $\mathcal{A}(z;E_{\mathrm{p,o}},F)$; in this case, each simulated GRB in Yonetoku parameter space is characterized by the parameters $\{L_{\mathrm{p}},E_{\mathrm{p},z_{\mathrm{g}}},f_{\gamma},E_{\mathrm{p,o}},z_{\mathrm{g}}\}$ and warrants a separate investigation. We follow a similar procedure outlined for the Amati relation when assigning the intrinisic parameters of the simulated GRBs, and dispersion according to $\sigma_{\log E_{{\rm p},z}}$ = 0.25 (Ghirlanda et al., 2005), which defines the Yonetoku uncertainty band $\mathcal{U_{\mathrm{Y}}}$. We take $a_{Y}$ to be 0.625 $\pm 0.032$, and $b_{Y}$ to be -30.22 $\pm 0.023$ taken from literature (Yonetoku et al., 2010) ²²2From Yonetoku et al. (2010), $$L_{p}=10^{52.43\pm 0.037}\times\left[\frac{E_{p}(1+z)}{355\,\text{keV}}\right]^{1.60\pm 0.082}$$ Thus, $a_{Y}$ = $\frac{1}{1.6}$, and $b_{Y}$ = $\log(355)-\frac{52.43}{1.6}$ . We sample the luminosities from a simple lognormal distribution $\log_{10}L_{\text{peak}}\sim\mathcal{N}(\mu=52.5,\sigma^{2}=1)$. Note that the choice of luminosity function does not alter the key results, because the functional form of $\mathcal{Y}(z;E_{\mathrm{p,o}},f_{\gamma})$—and hence the geometry of its intersection with the Yonetoku correlation—remains unchanged. Consequently, only the relative weighting of the parameter space is affected, not the overall behavior of solutions.

In this paper, we examine the feasibility of analytically obtaining pseudo-redshifts using two well-known phenomenological correlations of the prompt emission, which have become the de facto method for this purpose. With the aid of a synthetic catalogue of GRBs, we find this practice to be untenable for the following reasons:

• •

  Analytically solving for $z_{\rm i}$ from the best fit Amati relation always results in two solutions, which are both within a physical range (except in the case of $z_{\mathrm{g}}=z_{\mathrm{t}}$). The Amati relation can be practically well behaved over $z\in[0.1,10]$ if the Amati slope $a_{\mathrm{A}}\geq 0.67$, well above the uncertainty range of $a_{\mathrm{A}}$. Intrinsic scatter would not salvage the situation; if intrinsic scatter is added, there would either be (i) two solutions, (ii) one solution, or (iii) no solution. In the case of no solution, taking the point on $\mathcal{A}(z;E_{\mathrm{p,o}},F)$ closest to the mean Amati track would only yield a constant $z_{\mathrm{i}}$, as a corollary of Equation 10. In the case of a single solution, it is easy to see that in all cases $z_{\mathrm{i}}<z_{\mathrm{t}}$, and hovers around a constant value. Therefore, the Amati relation fails at a fundamental level and cannot be reliably used as a distance indicator

• •

  Although the Yonetoku relation results in a unique solution for $z_{\mathrm{i}}$, the intrinsic scatter leads to there being no solution for about 5% simulated GRBs and little predictive power of $z_{\mathrm{i}}$.

• •

  Incorporating the confidence areas $\mathcal{U_{\rm A}}$ and $\mathcal{U_{\rm Y}}$ would yield excessively large error bands for $z_{\mathrm{i}}$ for both the Amati and Yonetoku relations.

We have not simulated GRBs with uncertainty bands in $E_{\mathrm{p}}$ and $L_{\mathrm{p}}$, so in practice, we would expect even larger uncertainties in $z_{\mathrm{i}}$ if the uncertainties of fluence and peak energy are accounted for. In order to check whether the redshift distribution of the simulated GRB population will change the above analysis on the Yonetoku relation, we repeat the analysis with a different redshift distribution, a density evolved rate, as proposed in Lan et al, ie, $\psi_{\rm GRB}=\psi_{*}(z)(1+z)^{\delta}$, with $\delta=1.9$, which places more GRBs at higher redshifts Lan et al. (2019). In this case, the strength of the $z_{i}$ vs $z_{g}$ correlation decreases to $r=0.33$, and the curves of about 20% of LGRBs would not intersect the Yonetoku line, and hence have no solution. In the $z_{\rm i}$ vs. $z_{\rm g}$ plots shown in Figure 4, not only is the scatter around the $z=z_{\rm g}$ line quite large, but the confidence intervals for individual $z_{i}$ are also very wide—exceeding 2.5 for GRBs with $z_{\rm g}>1.5$. For LGRBs, the 1$\sigma$ confidence interval of $z_{i}$ (upper limit capped at 10) averages to 4.19, which is broader than the characteristic width ($\Delta z\sim 3$) of the metallicity-convolved GRB rate distribution. It is critical to emphasize again that this result is without considering for errors in measurements of fluence and peak energy. Such high uncertainties of the inferred $z_{\rm i}$ (significantly larger than the width of $z$ distribution of the population) calls into question the feasibility of any population studies which aim to infer distributions directly from pseudo-redshifts; for instance, it would be difficult to obtain any meaningful constraints on our understanding between the correlation of LGRBs and the cosmic star formation history, the characteristic delay time, or the luminosity function.

In this work, we have assumed a k-correction factor of 1; in reality, the curves of $\mathcal{A}(z;E_{\mathrm{p,o}},F)$ and $\mathcal{Y}(z;E_{\mathrm{p,o}},f_{\gamma})$ would be slightly modified. Zegarelli et al. (2022) found the k-correction distribution of GRBs detected by Fermi-GBM to be highly skewed towards 1, with a median value of 1.12 and a mean value of 1.27, and with a maximum value of an outlier at $\sim$ 4. Therefore, although in principle $k(z)$ would increase with $z$, the effect would be negligible due to the wide energy coverage of Fermi-GBM. In fact, Paul (2018) demonstrated that by assuming Fermi-GBM’s GRBs follow the Band function and averaging the low- and high-energy indices $\alpha$ and $\beta$, the average spectral shape detected by Fermi-GBM yields a k-correction below 1.5 even at $z=10$.